ON THE MOTION OF PENDULUMS. 75 The work of this pressure during the time dt corresponding to the element of surface dx dz, is equal to Hence the work exerted over a given portion of the surface is equal to 2fj,msc*dt ffda dz. In the absence of pressures AP2, AT3 at the surface, this work must be supplied at the expense of ms viva. Hence 4/j,m8c*dt ffda dz is the vis viva lost by friction, which agrees with the expression obtained in Art. 51, as will be seen on performing in the latter the integration with respect to y, the limits being y = 0 to y = oo .